Binary Converter

Enter the number, set the input base to decimal, and convert. Every other base is shown at once, so you get binary, hexadecimal and octal from a single conversion rather than one at a time.

Because a physical circuit reliably distinguishes two states, on and off, far better than it distinguishes ten. Every number, character and instruction inside a machine is ultimately a pattern of those two states, which is what base 2 describes.

Compactness. One hex digit is exactly four binary digits, so a byte is always two hex characters and long binary strings become short and readable. That is why colours are written as #FF8800, why memory addresses and byte dumps use hex, and why permission masks are quoted in it.

Each position is a power of two, doubling from right to left: 1, 2, 4, 8, 16 and so on. Add up the positions holding a 1. So 1011 is 8 plus 0 plus 2 plus 1, which is 11 in decimal.

Mostly through Unix file permissions, where each digit neatly encodes read, write and execute as three bits, which is how 755 and 644 came to be written that way. It was more widely used on older machines whose word sizes divided evenly by three.

It converts whole non-negative numbers between bases. Negative values in binary depend on a chosen representation such as two's complement and a fixed bit width, which the Binary Calculator handles instead.